Chaos

There are two groups of people in this world: those who love mathematics and those who detest it. Four out of ten Americans fall into the latter category and cite difficulty, uselessness and lack of creativity in math, among other things, as reasons for their distaste[1]. Many sentiments among this group are shared with Calvin [2].

I would argue that mathematics is not inherently negative, only woefully misunderstood. A misguided emphasis in the teaching of the subject has scared many people off. For years, students are bombarded with the complex language and rules of math, learning how multiplication works and what inequalities mean and that 8+2=10. Little time is spent discussing the true importance of math: how the language and rules transcend time and space to express the patterns of the universe, that inequalities describe how two objects compare and that addition combines groups, ideas or quantities to create something new. A celebration of discovery through math, rather than an expectation of correct answers at all costs, might cause more to agree with Lord Kelvin (or whomever is the appropriate originator) and “not imagine that mathematics is harsh and crabbed”.

Those on Kelvin’s side of the debate already believe in math as “the etherealization of the common sense.” Omnipresent mathematics is not invented so much as expressed in what Galileo dubbed the “language of the Universe.” British Mathematician G.H. Hardy said it best: “I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove … Are simply the notes of our observations”[3]. Mathematics ought not to be exclusively characterized as endlessly difficult problem sets, but as a tool (or more accurately a toolbox) for describing the workings of the universe. It may be taken as a means for answering the questions of philosophy, if we take philosophy as the most fundamental way of comprehending the world.

With mathematics properly contextualized, I want to enliven a small section of the discipline that is relatively young called chaos theory. Specifically, I want to examine the theory with a focus on the metaphysical problem of determinism. Lastly, I will consider a Christian perspective on divine action that is consistent with determinism and chaos theory and helps to bridge the two.

Making sense of chaos: a crash course

Professor of Philosophy Stephen Kellert describes chaos in short as “unstable aperiodic behavior in deterministic nonlinear dynamical systems”[4]. Kellert’s definition is generally accepted by scholars. Although like any concept bridging math, science and philosophy, the idea of ‘chaos’, and its legitimacy, is hotly debated. Given the already ambitious goals of my paper, I won’t delve into that gray area. Instead, throughout the paper I will simply view chaos theory as an exciting new discipline with great potential to propel current mathematical and philosophical thinking forward. To set the stage, let’s briefly discuss the history of chaos and unpack Kellert’s technical and somewhat intimidating definition.

Chaos was examined in conjunction with specific problems as early as the 1880s and appeared as an unknown error in later studies of nonlinear differential equations. It became a dedicated field of inquiry following the 1963 publication of a famous paper titled “Deterministic Nonperiodic Flow” by MIT Meteorologist and Mathematician Edward Lorenz. In the paper, Lorenz delineates his accidental discovery that computer simulations of a weather system exhibited a high level of sensitive dependence. In other words, a miniscule change in initial conditions would yield a drastically different outcome. He wrote, “one meteorologist remarked that if the theory were correct, one flap of a sea gull’s wings would be enough to alter the course of the weather forever. The controversy has not yet been settled, but the most recent evidence seems to favor the sea gulls”. In the years since Lorenz introduced this “butterfly effect,” researchers have developed a mathematical definition and have discovered that the phenomena effects macroscopic economic, biological and physical systems alike.

Kellert’s succinct definition tells us that while the behavior of many real-life systems can be modeled by equations and can result directly from initial conditions, chaos renders the systems unpredictable. Chaos does answer to some long-range order and is thus nonrandom. Put differently, it exhibits attributes such as feedback, self-similarity and geometric patterns, occurrences that are not random. But current methods of analysis cannot resolve these defining patterns and yield only what’s called the “strange attractor” of a chaotic system: the set of possible behaviors (think about this like the set of all possible outcomes of a given event). Thus, behavior appears to be random, since it inhabits some, seemingly randomly selected outcome in the set of all possible outcomes. This problem is incredibly complex in theory but becomes even more difficult when applied to natural, observable systems. The scientist is restricted by instrumentation and can only know the initial conditions of a system with limited precision. As an introductory science lecturer excitedly affirms somewhere far off in the distance, significant figures are incredibly relevant. Lorenz writes that “the present determines the future, but the approximate present does not approximately determine the future”[5]. As an aside, the classic example of a chaotic system is that of a double pendulum – I recommend you explore Youtube videos or its Wikipedia page.

Chaos + philosophy

Chaos theory certainly creates some interesting challenges for mathematics and for fields that rely on mathematics. Yet if chaos is a trial for mathematics, it is a boon for the metaphysical proposition of determinism. Determinism is the idea that “given the way things are at a time t, the way things go thereafter is fixed as a matter of natural law”[6]. If chaos occurs naturally in deterministic systems and not only theoretically, it must affect reality-describing metaphysics. To show that naturally-occurring chaos is a stronger case for determinism, two questions must be answered. First, are chaotic deterministic systems natural or are they simply a byproduct of mathematical theory? Second, if such systems are natural, do they strengthen the case for determinism or for its opposite, indeterminism?

It is understandably tricky to obtain empirical evidence for chaos given its unpredictability. The math is incredibly difficult, the validity of a faithful model assumption is necessarily questionable (a faithful model assumption says that if a model accurately describes its target system, the model will become ever-more accurate as it is further refined) and the precision of instrumentation is limiting. Curiously, strong evidence for natural chaos has been used to understand a variety of problems such as the chaotic attributes of a dripping faucet. Certain confirmable higher-order mathematical models exhibit chaotic stability, the existence of natural chaos, among their corresponding strange attractors. There are no strong counterarguments save skepticism which, while healthy to maintain in an investigative field, does not outweigh reasonably founded conclusions.

Chaotic systems are now relevant to the metaphysical discussion and must thus be characterized as most consistent with either determinism or indeterminism. The view taken by well-known philosopher of science Karl Popper is that the unpredictability of chaotic systems implies indeterminism. However, Popper takes uncertainty to be inherent to reality (ontological), when it is actually just a governor on the tools of science and math (epistemic)[7]. Popper sells science short. The fact that mathematics cannot currently discern higher-order patterns of chaos in no way means that they are nonexistent; furthermore, there is evidence of such patterns! Influential French scholar Pierre-Simon Laplace makes the distinction: “We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes”[8]. Therefore it is clear that “the apparent randomness of even simple natural dynamical systems [may] be brought under the umbrella of determinism”[9]. However, the very epistemic issues raised by chaos which strengthen determinism’s case also limit its ability to describe reality itself, rather than a perception of reality. Mathematics may evolve to encompass chaos, but uncertainty will necessarily set in with initial condition values that have infinite decimal representation (in other words, numbers as intellectually grasped must have some degree of imprecision). This is where propositions for divine action begin. Might an omnipotent God who is himself unbridled with the laws of science choose to interact with the world in a deterministic manner?

A proposition

Bolstered by chaos theory, determinism tells us that all action in the world is a series of cause and effect events; external agents can do nothing to affect event outcomes in this “closed causal web”. This stands in contrast with a Christian view that seems to be necessarily indeterminist in describing the divine actions of sustenance, governance and cooperation. It would appear that the order of the world must be open to allow for God to preserve the world and “make clouds rise from the ends of the earth, send lightning with the rain and bring out wind from his storehouses” and induce phenomena such as the great flood and Jesus’ miracles. (Psalm 135:6- 7, New International Version). It is therefore prudent for the ‘rational Christian’ to seek a perspective that is simultaneously consistent with biblical teaching and with chaos and its metaphysical implications. A solution to the apparent incongruity is presented after arguments that would leave either the philosopher or the theologian dissatisfied.

Many theologians are quick to view the existence of chaos as grounds for dismissing determinism, allowing God to act as he pleases. John Polkinghorne, a physicist and philosopher turned Anglican priest, writes that “the apparently deterministic proves to be intrinsically unpredictable. It is suggested that the natural interpretation of this exquisite sensitivity is to treat it, not merely as an epistemological barrier, but as an indication of the ontological openness of the world of complex dynamical systems”[10]. While Polkinghorne makes an incredibly interesting case in his paper “The Metaphysics of Divine Action,” this argument is unsatisfactory to the metaphysician because it interprets chaos as being indicative of indeterminism. As we established in the previous section, this is a logical jump. The Stanford Encyclopedia of Philosophy cautions that “if Polkinghorne’s suggestions amount to simply viewing the world as if chaos harbors indeterminism, then it seems that this suggestion doesn’t yield the kind of divine action he seeks”[11]. Chaos does not serve as a back door to undermine scientific and philosophical reasoning, so another explanation must be put forth.

Similarly, a number of theories explain chaotic uncertainty under a deterministic framework but do not include divine influence, or include it in a manner that poorly describes the God of Christianity. Some, like Ilya Prigogine and his Brussels-Austin Group, suggest an alternative yet unsatisfactory approach to physics that more fully explains the situation at hand: “arrangements of particles are fundamental explanatory elements and not individual particles and trajectories”[12]. Under such a framework, the uncertainty and probability of chaotic systems become ontologically fundamental such that determinism still holds. This is different from Karl Popper’s argument because it defines a new approach to physics instead of creating an inconsistency in the old system. Unfortunately, this remains unproven speculation and the theologian is left wondering where God fits into the model. Perspectives such as deism account for “a God,” but one that is distant and limited in power. Deism deterministically holds that God set the world in motion (set the initial conditions for the entire cause and effect chain of history) and then backed off like you or I would wind a clock. This view renders His mercy in the Old Testament, prayer, and the reassuring tenet that “the Lord …Will never leave you or forsake you” (Deuteronomy 31:6) sadly irrelevant. While the above theories and others like them are philosophically consistent, they lack an essential theological robustness.

We begin the search for a “middle ground” by revisiting Laplace. Again, he states that “an intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.” It is no stretch to imagine that the Christian God by whom “the very hairs of your head are all numbered” possesses such an intellect. (Luke 12:7). Yet it feels improper to describe God as an incredibly powerful calculator; a calculator seems unlikely to offer himself as a living sacrifice for humanity, the central tenet of Christianity, or to “take great delight in you, quiet you with His love [and] rejoice over you with singing.” (Zephaniah 3:17) To break this down further, it suggests the nature of God is one that presently sustains and interacts with all entities of the world in a manner that cooperates with their characteristic limitations.

The view that God is omnipotent (all-powerful) implies that he is not obligated to act deterministically. Yet it is postulated that he does, working through the laws of nature governing earthly activity. He acts according to a world which He set up with structure and predictability, so that humans, striving to glorify God and enjoy Him forever, may do so against “a background of law-like processes so that the effects of one’s actions can be predicted”[13]. This story echoes a central Biblical theme: God does not do something because he must, but does something because he loves. Imagine living in a world with even some degree of objective indeterminism. If everyday activity A does not always give rise to positive outcome B, what fear there would be if some of the time A gave rise to catastrophic outcome C! Not only is this contrary to the Christian teaching of a God who “knows the plans [he] has for [us]” but it also contradicts a wealth of mathematical models that show, in spite of chaos, a fundamental orderliness. (Jeremiah 29:11) If determinism is an umbrella, God’s actions are the weather around the umbrella necessitating how it is used.

The means by which this divine action is caused falls some what outside of this discussion, though an intriguing hypothesis is set forth by American Philosopher Nancey Murphey in her essay “Divine Action in the Natural Order.” Murphey explores the quantum world as the mechanism for God’s interaction. She argues that He engineers the precise set of initial conditions, from a myriad of valid quantum states, to yield a particular event. The world remains deterministic because all actions occur starting within the natural possibilities of behavior for atoms, propagating to the macroscopic world in a “bottom up” manner. One somehow possessing the intellect set forth by Laplace and sufficient information could thus model the event. Such a theory is also theologically consistent as it acknowledges a God working in and through all things, accounts for “miracles” in the envelope of chaotic and quantum outcomes and allows humans to “know, love and serve him” freely, understanding the results in a well-defined deterministic framework. Murphey herself acknowledges limitationslike dealing with the specificity and complexity of quantam theory- to this proposal, yet it stands out amongst others for its adherence to both scientific and theological prerequisites[14].

Conclusions

It should be remembered that the ideas presented here are predicated entirely on a particular philosophical view: that the world is deterministic and could be entirely described with laws and equations if they were somehow known. Anyone justifiably subscribing to an alternative view would therefore disregard this discussion as irrelevant. Similarly, details of the game change when delving into the quantum world. But with determinism having been demonstrated as an increasingly plausible framework for understanding the macroscopic world, the conversation is worth having. Just as this basis remains disputable, the relationship between divine action and chaos-influenced reality will forever be debated. The book of Isaiah reminds that “as the heavens are higher than the earth, so are [God’s] ways higher than your ways and [his] thoughts than your thoughts” (Isaiah 55:8- 9). This article does not seek to close the book on the issue, as a full understanding can never be reached. Instead, it seeks to turn to the next chapter as mathematics evolves and as Christian theology integrates that progress with its interpretation of God and existence. This article has successfully demonstrated, if anything else, that humanity is far from a universal understanding. The epistemological barriers necessitated by chaos appear as if they will keep us forever in the dark. Perhaps the only certainty here is that much remains mysterious. Fear, accompanied by big questions about purpose and fate, often follows irresolution and misunderstanding. The relationship of 4 in 10 Americans to mathematics has proven as much. While science continues the search for a light switch, there remains a more fundamental comfort amidst incalculable chaos: that “[God]is before all things, and in him all things hold together.” (Colossians 1:17)

Note: Readers with further interest in this and similar topics are highly encouraged to read some of the essays in “Chaos and Complexity: Scientific Perspectives on Divine Action.”

Taylor Parra is a senior double majoring in mechanical engineering and applied mathematics, with a minor in materials science. He recently returned from studying abroad in Glasgow, Scotland and is training to be a military pilot.